W05 Notes

Hydrostatic force

Videos, Organic Chemistry Tutor

01 Theory

The pressure in a liquid is a function of the depth alone. This is a fundamental fact about liquids.

Pressure function

The fluid pressure in a liquid is a function of depth:

In SI units:

The pressure of a fluid acts upon any surface in the fluid by exerting a force perpendicular to the surface. Force is pressure times area. If the pressure varies across the surface, the total force must be calculated using an integral to add up differing contributions of force on each portion of the surface.
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Fluid force on submerged plate

Total fluid force on plate:

Use for top of plate (shallow) and for bottom of plate (deepest).
Have at the surface.

  • This formula assumes the plate is oriented straight vertically, not slanting.
  • Note that increases with depth, reverse from the usual -axis.

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02 Illustration

Fluid force on a triangular plate

15 - Fluid force on a triangular plate

Find the total force on the submerged vertical plate with the following shape: Equilateral triangle, sides , top vertex at the surface, liquid is oil with density .
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Solution

  1. Establish coordinate system: height increases going down.
  2. Compute width function .
    • Drop a perpendicular from top vertex to the base.
    • Pythagorean Theorem: vertical height is .
    • Similar triangles: ratio must equal ratio .
    • Solve for :
  3. Write integral using width function.
    • Bounds: shallowest: ; deepest: .
    • Integral formula:
  4. Compute integral.
    • Simplify constants:
    • Compute integral without constants:
    • Combine for the final answer:

Moments and center of mass

Videos, Math Dr. Bob:

03 Theory

Moment

The moment of a region to an axis is the total (integral) of mass times distance to that axis:

“Moment to :”

“Moment to :”

  • Notice the swap in letters! has integral with while has integral with .
    • Because needs distance to -axis.
  • Notice that if you remove or factors from the integrands, the integrals give total mass .

These formulas are obtained by slicing the region into rectangular strips that are parallel to the axis in question.

The area per strip is then:

  • — region under
  • — region between and
  • — region ‘under’
  • — region between and

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The idea of moment is related to:

  • Torque balance and angular inertia
  • Center of mass

The center of mass (CoM) of a solid body is a single point with two important properties:

  1. “average position” of the body
    • The average position determines an effective center of dynamics. For example, gravity acting on every bit of mass of a rigid body acts the same as a force on the CoM alone.
  2. “balance point” of the body
    • The net torque (rotational force) about the CoM, generated by a force distributed over the body’s mass, equivalently a force on the CoM, is zero.

Note:

  • When the body has uniform density, then the CoM is also called the centroid.
Center of mass from moments

Coordinates of the CoM:

Here and are the moments and is the total mass of the body.

Center of mass from moments - explanation

Notice how these formulas work. The total mass is always . The moment to (for example) is . Dividing these:
⨠⨠⨠⨠⨠⨠where .

In other words, through the formula , we find that is the average value of over the region with area .

04 Illustration

CoM of a parabolic plate

17 - CoM of a parabolic plate

Find the CoM of the region depicted:
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Solution

  1. Compute the total mass.
    • Area under the curve with density factor : ⨠⨠⨠⨠
  2. Compute .
    • Formula:
    • Interpret and calculate: ⨠⨠⨠⨠
  3. Compute .
    • Formula:
    • Width of horizontal strips between the curves:
    • Interpret :
    • Plug data into integral: ⨠⨠
    • Calculate integral: ⨠⨠⨠⨠
  4. Compute CoM coordinates from moments.
    • CoM formulas:
    • Insert data: ⨠⨠⨠⨠
    • Therefore CoM is located at .

05 Theory

A downside of the technique above is that to find we needed to convert the region into functions in . This would be hard to do if the region was given as the area under a curve but cannot be found analytically. An alternative formula that works in this situation.

Midpoint of strips for opposite variables

When the region lies between and , we can find with an -integral:

When the region lies between and , we can find with a -integral:

  • Use or for regions “under a curve” or .

The idea for these formulas is to treat each vertical strip as a point concentrated at the CoM of the vertical strip itself.

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The height to this midpoint is , and the area of the strip is , so the integral becomes .

Midpoint of strips formula - full explanation
  • If the strip is located at some , with values from up to , then:
  • The area of the strip is . So the integral formula for can be recast: ⨠⨠⨠⨠
  • If the vertical strips are between and , then the midpoints of the strips are given by the ‘average’ function:
  • The height of each strip is , so .
  • Putting this together: ⨠⨠⨠⨠

06 Illustration

Computing CoM using only vertical strips

18 - Computing CoM using only vertical strips

Find the CoM of the region:
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Solution

  1. Compute the total mass .
    • Area under the curve times density :
  2. Compute using vertical strips.
    • Plug into formula: ⨠⨠
    • Integration by parts.
      • Set , ; then , .
      • IBP formula:
      • Plug in data:
      • Evaluate: ⨠⨠
  3. Compute also using vertical strips.
    • Plug and into formula: ⨠⨠
    • Integration by ‘power to frequency conversion’:
      • Use :
      • Integrate: ⨠⨠
  4. Compute CoM.
    • CoM via moment formulas:
    • Plug in data: ⨠⨠
    • Plug in data: ⨠⨠
    • CoM is given by .

CoM of region between line and parabola

19 - CoM of region between line and parabola

Compute the CoM of the region below and above with .

Solution

  1. Name the functions: (lower) and (upper) over .
  2. Compute the mass .
    • Area between curves times density: ⨠⨠
  3. Compute using vertical strips.
    • Plug into formula:
  4. Compute also using vertical strips.
    • Plug into formula: ⨠⨠
  5. Compute CoM.
    • Using CoM via moment formulas: ⨠⨠⨠⨠
    • CoM is given by .

07 Theory

Two useful techniques for calculating moments and (thereby) CoMs:

  • Additivity principle
  • Symmetry

Additivity says that you can add moments of parts of a region to get the total moment of the region (to a given axis).

A symmetry principle is that if a region is mirror symmetric across some line, then the CoM must lie on that line.

08 Illustration

Center of mass using moments and symmetry

20 - Center of mass using moments and symmetry

Find the center of mass of the region below.
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Solution

  1. Symmetry: CoM on -axis.
    • Because the region is symmetric in the -axis, the CoM must lie on that axis.
    • Therefore .
  2. Additivity of moments.
    • Write for the total -moment (distance measured to the -axis from above).
    • Write and for the -moments of the triangle and circle.
    • Additivity of moments equation:
  3. Find moment of the circle .
    • By symmetry we know .
    • By symmetry we know .
    • Area of circle with is , so total mass is .
    • Centroid-from-moments equation:
    • Solve the equation for .
      • Solve: ⨠⨠⨠⨠
  4. Find moment of the triangle using integral formula
    • Similar triangles:
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    • Similarity equation: ⨠⨠
    • Integral formula:
    • Perform integral: ⨠⨠
    • Conclude:
  5. Apply additivity.
    • Additivity formula:
  6. Total mass of region.
    • Area of circle is .
    • Area of triangle is .
    • Thus .
  7. Compute center of mass from total and total .
    • We have and .
    • Plug into formula:
  8. Final answer is .

Center of mass - two part region

21 - Center of mass - two part region

Find the center of mass of the region which combines a rectangle and triangle (as in the figure) by computing separate moments. What are those separate moments? Assume the mass density is .
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Solution

  1. By symmetry, the center of mass of the rectangle is located at .
    • Thus and .
  2. Find moments of the rectangle.
    • Total mass of rectangle .
    • Apply moment relation: ⨠⨠⨠⨠
  3. Find moments of the triangle.
    • Area of vertical slice .
    • Distance from -axis .
    • Total integral:
    • Total integral:
  4. Add up total moments.
    • General formulas: and
    • Plug in data: and
  5. Find center of mass from moments.
    • Total mass of triangle .
    • Total combined mass .
    • Apply moment relation:
    • Therefore, center of mass is .